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Theorem ltexpri 7603

Description: Proposition 9-3.5(iv) of [Gleason] p. 123. (Contributed by NM, 13-May-1996.) (Revised by Mario Carneiro, 14-Jun-2013.)

**Assertion**
Ref	Expression
ltexpri	⊢ (𝐴<_P 𝐵 → ∃𝑥 ∈ P (𝐴 +_P 𝑥) = 𝐵)

Distinct variable groups: 𝑥,𝐴 𝑥,𝐵

Proof of Theorem ltexpri Dummy variables 𝑦 𝑧 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 110	. . . . . . . 8 ⊢ ((𝑦 = 𝑢 ∧ 𝑧 = 𝑣) → 𝑧 = 𝑣)
2	1	eleq1d 2246	. . . . . . 7 ⊢ ((𝑦 = 𝑢 ∧ 𝑧 = 𝑣) → (𝑧 ∈ (2^nd ‘𝐴) ↔ 𝑣 ∈ (2^nd ‘𝐴)))
3		simpl 109	. . . . . . . . 9 ⊢ ((𝑦 = 𝑢 ∧ 𝑧 = 𝑣) → 𝑦 = 𝑢)
4	1, 3	oveq12d 5887	. . . . . . . 8 ⊢ ((𝑦 = 𝑢 ∧ 𝑧 = 𝑣) → (𝑧 +_Q 𝑦) = (𝑣 +_Q 𝑢))
5	4	eleq1d 2246	. . . . . . 7 ⊢ ((𝑦 = 𝑢 ∧ 𝑧 = 𝑣) → ((𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵) ↔ (𝑣 +_Q 𝑢) ∈ (1^st ‘𝐵)))
6	2, 5	anbi12d 473	. . . . . 6 ⊢ ((𝑦 = 𝑢 ∧ 𝑧 = 𝑣) → ((𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵)) ↔ (𝑣 ∈ (2^nd ‘𝐴) ∧ (𝑣 +_Q 𝑢) ∈ (1^st ‘𝐵))))
7	6	cbvexdva 1929	. . . . 5 ⊢ (𝑦 = 𝑢 → (∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵)) ↔ ∃𝑣(𝑣 ∈ (2^nd ‘𝐴) ∧ (𝑣 +_Q 𝑢) ∈ (1^st ‘𝐵))))
8	7	cbvrabv 2736	. . . 4 ⊢ {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))} = {𝑢 ∈ Q ∣ ∃𝑣(𝑣 ∈ (2^nd ‘𝐴) ∧ (𝑣 +_Q 𝑢) ∈ (1^st ‘𝐵))}
9	1	eleq1d 2246	. . . . . . 7 ⊢ ((𝑦 = 𝑢 ∧ 𝑧 = 𝑣) → (𝑧 ∈ (1^st ‘𝐴) ↔ 𝑣 ∈ (1^st ‘𝐴)))
10	4	eleq1d 2246	. . . . . . 7 ⊢ ((𝑦 = 𝑢 ∧ 𝑧 = 𝑣) → ((𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵) ↔ (𝑣 +_Q 𝑢) ∈ (2^nd ‘𝐵)))
11	9, 10	anbi12d 473	. . . . . 6 ⊢ ((𝑦 = 𝑢 ∧ 𝑧 = 𝑣) → ((𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵)) ↔ (𝑣 ∈ (1^st ‘𝐴) ∧ (𝑣 +_Q 𝑢) ∈ (2^nd ‘𝐵))))
12	11	cbvexdva 1929	. . . . 5 ⊢ (𝑦 = 𝑢 → (∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵)) ↔ ∃𝑣(𝑣 ∈ (1^st ‘𝐴) ∧ (𝑣 +_Q 𝑢) ∈ (2^nd ‘𝐵))))
13	12	cbvrabv 2736	. . . 4 ⊢ {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))} = {𝑢 ∈ Q ∣ ∃𝑣(𝑣 ∈ (1^st ‘𝐴) ∧ (𝑣 +_Q 𝑢) ∈ (2^nd ‘𝐵))}
14	8, 13	opeq12i 3781	. . 3 ⊢ ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩ = ⟨{𝑢 ∈ Q ∣ ∃𝑣(𝑣 ∈ (2^nd ‘𝐴) ∧ (𝑣 +_Q 𝑢) ∈ (1^st ‘𝐵))}, {𝑢 ∈ Q ∣ ∃𝑣(𝑣 ∈ (1^st ‘𝐴) ∧ (𝑣 +_Q 𝑢) ∈ (2^nd ‘𝐵))}⟩
15	14	ltexprlempr 7598	. 2 ⊢ (𝐴<_P 𝐵 → ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩ ∈ P)
16	14	ltexprlemfl 7599	. . . 4 ⊢ (𝐴<_P 𝐵 → (1^st ‘(𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩)) ⊆ (1^st ‘𝐵))
17	14	ltexprlemrl 7600	. . . 4 ⊢ (𝐴<_P 𝐵 → (1^st ‘𝐵) ⊆ (1^st ‘(𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩)))
18	16, 17	eqssd 3172	. . 3 ⊢ (𝐴<_P 𝐵 → (1^st ‘(𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩)) = (1^st ‘𝐵))
19	14	ltexprlemfu 7601	. . . 4 ⊢ (𝐴<_P 𝐵 → (2^nd ‘(𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩)) ⊆ (2^nd ‘𝐵))
20	14	ltexprlemru 7602	. . . 4 ⊢ (𝐴<_P 𝐵 → (2^nd ‘𝐵) ⊆ (2^nd ‘(𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩)))
21	19, 20	eqssd 3172	. . 3 ⊢ (𝐴<_P 𝐵 → (2^nd ‘(𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩)) = (2^nd ‘𝐵))
22		ltrelpr 7495	. . . . . . 7 ⊢ <_P ⊆ (P × P)
23	22	brel 4675	. . . . . 6 ⊢ (𝐴<_P 𝐵 → (𝐴 ∈ P ∧ 𝐵 ∈ P))
24	23	simpld 112	. . . . 5 ⊢ (𝐴<_P 𝐵 → 𝐴 ∈ P)
25		addclpr 7527	. . . . 5 ⊢ ((𝐴 ∈ P ∧ ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩ ∈ P) → (𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩) ∈ P)
26	24, 15, 25	syl2anc 411	. . . 4 ⊢ (𝐴<_P 𝐵 → (𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩) ∈ P)
27	23	simprd 114	. . . 4 ⊢ (𝐴<_P 𝐵 → 𝐵 ∈ P)
28		preqlu 7462	. . . 4 ⊢ (((𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩) ∈ P ∧ 𝐵 ∈ P) → ((𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩) = 𝐵 ↔ ((1^st ‘(𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩)) = (1^st ‘𝐵) ∧ (2^nd ‘(𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩)) = (2^nd ‘𝐵))))
29	26, 27, 28	syl2anc 411	. . 3 ⊢ (𝐴<_P 𝐵 → ((𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩) = 𝐵 ↔ ((1^st ‘(𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩)) = (1^st ‘𝐵) ∧ (2^nd ‘(𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩)) = (2^nd ‘𝐵))))
30	18, 21, 29	mpbir2and 944	. 2 ⊢ (𝐴<_P 𝐵 → (𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩) = 𝐵)
31		oveq2 5877	. . . 4 ⊢ (𝑥 = ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩ → (𝐴 +_P 𝑥) = (𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩))
32	31	eqeq1d 2186	. . 3 ⊢ (𝑥 = ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩ → ((𝐴 +_P 𝑥) = 𝐵 ↔ (𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩) = 𝐵))
33	32	rspcev 2841	. 2 ⊢ ((⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩ ∈ P ∧ (𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩) = 𝐵) → ∃𝑥 ∈ P (𝐴 +_P 𝑥) = 𝐵)
34	15, 30, 33	syl2anc 411	1 ⊢ (𝐴<_P 𝐵 → ∃𝑥 ∈ P (𝐴 +_P 𝑥) = 𝐵)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1353 ∃wex 1492 ∈ wcel 2148 ∃wrex 2456 {crab 2459 ⟨cop 3594 class class class wbr 4000 ‘cfv 5212 (class class class)co 5869 1^st c1st 6133 2^nd c2nd 6134 Qcnq 7270 +_Q cplq 7272 Pcnp 7281 +_P cpp 7283 <_P cltp 7285

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4115 ax-sep 4118 ax-nul 4126 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533 ax-iinf 4584

This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-tr 4099 df-eprel 4286 df-id 4290 df-po 4293 df-iso 4294 df-iord 4363 df-on 4365 df-suc 4368 df-iom 4587 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-res 4635 df-ima 4636 df-iota 5174 df-fun 5214 df-fn 5215 df-f 5216 df-f1 5217 df-fo 5218 df-f1o 5219 df-fv 5220 df-ov 5872 df-oprab 5873 df-mpo 5874 df-1st 6135 df-2nd 6136 df-recs 6300 df-irdg 6365 df-1o 6411 df-2o 6412 df-oadd 6415 df-omul 6416 df-er 6529 df-ec 6531 df-qs 6535 df-ni 7294 df-pli 7295 df-mi 7296 df-lti 7297 df-plpq 7334 df-mpq 7335 df-enq 7337 df-nqqs 7338 df-plqqs 7339 df-mqqs 7340 df-1nqqs 7341 df-rq 7342 df-ltnqqs 7343 df-enq0 7414 df-nq0 7415 df-0nq0 7416 df-plq0 7417 df-mq0 7418 df-inp 7456 df-iplp 7458 df-iltp 7460

This theorem is referenced by: lteupri 7607 ltaprlem 7608 ltaprg 7609 ltmprr 7632 recexgt0sr 7763 mulgt0sr 7768 map2psrprg 7795

W3C validator